One of the most active and continuing subjects in matrix theory during the last century, is the study of those linear operators on matrices that leave certain properties or relations of subsets invariant. Such questions are usually called ”Linear Preserver Problems”. The earliest papers in our reference list are Frobenius(1897) and Kantor(1897). Since much effort has been devoted to this type of problem, there have been several excellent survey papers. For survey of these types of problems, we refer to the article of Song([11]) and the papers in [10]. The specified frame of problems is of interest both for matrices with entries from a field and for matrices with entries from an arbitrary semiring such as Boolean algebra, nonnegative integers, and fuzzy sets. It is necessary to note that there are several rank functions over a semiring that are analogues of the classical function of the matrix rank over a field. Detailed research and self-contained information about rank functions over semirings can be found in [1, 11].
There are some results on the inequalities for the rank function of matrices([1, 2, 3, 4]).
Beasley and Guterman ([1]) investigated the rank inequalities of matrices over semirings. And they characterized the equality cases for some rank inequalities in [2]. The investigation of linear preserver problems of extreme cases of the rank inequalities of matrices over fields was obtained in [4]. The structure of matrix varieties which arise as extremal cases in the inequalities is far from being understood over fields, as well as semirings. A usual way to generate elements of such a variety is to find a matrix pairs which belongs to it and to act on this set by various linear operators that preserve this variety. Song and his colleagues ([3]) characterized the linear operators that preserve the extreme cases of column rank inequalities over semirings.
There are some results on the linear operators that preserve term rank([7, 8]) and zero-term rank([5]). But in these papers, the authors studied the term rank and zero-term rank function themselves.
In this thesis, we characterize linear operators that preserve the sets of matrix pairs which satisfy extreme cases for the term rank inequalities and zero-term rank inequalities for the
product of matrices over fuzzy semirings.
Definition 1.1. ([3]) A semiring S consists of a set S and two binary operations, addition and
multiplication, such that:
• S is an Abelian monoid under addition (identity denoted by 0);
• S is a semigroup under multiplication (identity, if any, denoted by 1);
• multiplication is distributive over addition on both sides;
• s0 = 0s = 0 for all s ∈ S.
Definition 1.2. ([3]) A semiring is called antinegative if the zero element is the only element with an additive inverse.
Definition 1.3. ([5]) The Boolean semiring consists of the set B = {0, 1} equipped with two binary operations, addition and multiplication. The operations are defined as usual except that 1 + 1 = 1.
Definition 1.4. ([1]) A semiring is called chain if the set S is totally ordered with universal lower and upper bounds and the operations are defined by a + b = max{a, b} and a · b = min{a, b}.
It is straightforward to see that any chain semiring is commutative and antinegative.
Throughout we assume that m ≤ n. The matrix Inis the n × n identity matrix, Jm,n is the m × n matrix of all ones, Om,n is the m × n zero matrix. We omit the subscripts when the order is obvious from the context and we write I, J , and O, respectively. The matrix Ei,j, called a cell, denotes the matrix with exactly one nonzero entry, that being a one in the (i, j) entry. Let Ridenote the matrix whose ithrow is all ones and is zero elsewhere, and Cjdenote the matrix whose jthcolumn is all ones and is zero elsewhere. We let |A| denote the number of nonzero entries in the matrix A.
Let Mm,n(S) denote the set of m × n matrices with entries from the semiring S. If m = n, we use the notation Mn(S) insteed of Mm,n(S).
Definition 1.5. ([12]) Let R be the field of reals, let F ={α ∈ R | 0 ≤ α ≤ 1} denote a subset of reals. Define a + b = max{a, b} and a · b = min{a, b} for all a,b in F . Then (F , +, ·) is called a fuzzy semiring.
Let Mm,n(F) denote the set of all m × n matrices with entries in a fuzzy semiring F. We call a matrix in Mm,n(F) as a fuzzy matrix.
Definition 1.6. ([4]) A line of a matrix A is a row or a column of the matrix A.
Definition 1.7. ([7]) A matrix A ∈ Mm,n(F) has term rank k (t(A) = k) if the least number of lines needed to include all nonzero elements of A is equal to k. Let us denote by c(A) the least number of columns needed to include all nonzero elements of A and by r(A) the least number of rows needed to include all nonzero elements of A.
Definition 1.8. ([5]) A matrix A ∈ Mm,n(F) has zero-term rank k (z(A) = k) if the least number of lines needed to include all zero elements of A is equal to k.
Example 1.9. Let positive integer such that such a factorization exists. By definition the only matrix with factor rank equal to 0 is the zero matrix, O.
If S is a subsemiring of a certain field then there is a usual rank function ρ(A) for any matrix A ∈ Mm,n(S). It is easy to see that these functions are not equal in general but the inequality rank(A) ≥ ρ(A) always holds.
Example 1.11. Consider Z+, the set of nonnegative integers. The semiring Z+is embedded
in the real field R. Then the matrix
has different values as, where rank(A)=3 and ρ(A)=2.
Definition 1.12. ([2]) Let F be a fuzzy semiring. An operator T : Mm,n(F) → Mm,n(F) is called linear if T (X + Y ) = T (X) + T (Y ) and T (αX) = αT (X) for all X, Y ∈ Mm,n(F), α ∈ F.
Definition 1.13. ([3]) We say an operator, T , preserves a set P if X ∈ P implies that T (X) ∈ P, or, if (X, Y ) ∈ P implies that (T (X), T (Y )) ∈ P when P is a set of ordered pairs.
Definition 1.14. ([7]) The matrix X ◦ Y denotes the Hadamard or Schur product, i.e., the (i, j) entry of X ◦ Y is xi,jyi,j.
Definition 1.15. ([7]) An operator T is called a (P, Q, B)-operator if there exist permutation matrices P and Q, and a matrix B with no zero entries, such that T (X) = P (X ◦ B)Q for all X ∈ Mm,n(F), or, if m = n, T (X) = P (X ◦ B)tQ for all X ∈ Mm,n(F). The operator T (X) = P (X · B)Q is called nontransposing (P, Q, B)-operator. A (P, Q, B)-operator is called a (P, Q)-operator if B = J , the matrix of all ones.
It was shown in [2, 4, 9] that linear preserves for extremal cases of classical matrix inequal-ities over fields are types of (P, Q)-operators where P and Q are arbitrary invertible matrices.
On the other side, linear preservers for various rank functions over semirings have been the object of much study during the last years, see for example [6, 7, 8, 10], in particular term rank and zero term rank were investigated in the last few years, see for example [5].
Definition 1.16. ([5]) We say that the matrix A dominates the matrix B if and only if bi,j 6= 0 implies that ai,j 6= 0, and we write A ≥ B or B ≤ A.
Definition 1.17. If A and B are matrices and A ≥ B we let A\B denote the matrix C where
ci,j =
The behaviour of the function ρ with respect to matrix multiplication and addition is given by the following inequalities:
Sylvester’s laws:
ρ(A) + ρ(B) − n ≤ ρ(AB) ≤ min{ρ(A), ρ(B)},
and the Frobenius inequality:
ρ(AB) + ρ(BC) ≤ ρ(ABC) + ρ(B),
where A, B, C are conformal matrices with coefficients from a field.